A combination is the number of ways a given number of objects can be selected from a group when the order does not matter (unlike a permutation, in which order does matter). If k objects are selected from a group of n members, the formula for the combination (which can be read as "n choose k") is

$ {n \choose k} = \frac{n!}{k! (n-k)!} $

Combinations are important in probability as well as to the binomial theorem. All the possible combinations of an integer n make up the nth row of Pascal's triangle.


How many different groups of three can be chosen from five people?

Since order is not important, we can use the formula for a combination.

$ {5 \choose 3} = \frac{5!}{3! (5-3)!} = \frac{5!}{3! \cdot (2)!} = \frac{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}{3 \cdot 2 \cdot 1 \cdot (2 \cdot 1)} = \frac{5 \cdot 4}{2} = 10 $
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